This exercise shows that the hypotheses of Clairauts Theorem are needed. Let for (x, y) = (0,0)

This exercise shows that the hypotheses of Clairaut’s Theorem are needed. Let

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for (x, y) = (0,0) and f(0, 0) = 0. (a) Verify for (x, y) = (0,0): f(x, y) = xy- fx(x, y) = fy(x, y) = x - y x + y y(x4 + 4xy-y4) (x + y) x(x - 4xy - y) (x + y) (b) Use the limit definition of the partial derivative to show that fx(0,0) = f(0,0) = 0 and that fyx(0, 0) and fxy (0,0) both exist but are not equal. (c) Show that for (x, y) # (0,0): fxy(x, y) = fyx (x, y) = x6 + 9x^y - 9xy4y6 (x + y) Show that fry is not continuous at (0,0). Hint: Show that lim fry(h, 0) # lim fry(0, h). h0 h0 (d) Explain why the result of part (b) does not contradict Clairaut's Theorem.